{
  "abstract": "We prove uniform-in-$N$ partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel--Riesz kernels on $\\mathbb R^d$ in the locally square-integrable range $0\\le s<d/2$. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal $N^{-1}$ normalized relative-entropy rate for the three-dimensional Coulomb flow; and an $N^{-1/2}$ Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman--Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00867",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhenfu Wang"
    },
    {
      "name": "Xianliang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/719v659fvmesm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00867",
  "fulltext_url": null,
  "id": "719v659fvmesm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00867",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7206148801",
  "pdf_url": "https://arxiv.org/pdf/2609.00867",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/719v659fvmesm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "2609.00867",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00867"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Uniform Partition-Function Estimates for Coulomb Modulated Energy at All Positive Temperatures",
  "title_text": "Uniform Partition-Function Estimates for Coulomb Modulated Energy at All Positive Temperatures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/719v659fvmesm",
  "venue": "arXiv",
  "versions": []
}